Independent solution

How to solve this Uniform Distribution question

Setup

Setup

Let R denote the radial distance and A the resulting area.

R∼Uniform⁡(0,3)R\sim\operatorname{Uniform}(0,3)
A=πR2A=\pi R^2

Model

Model

Scaling by pi contributes pi squared to the variance, so the required inputs are the second and fourth radius moments.

Var⁡(A)=π2(E⁡[R4]−(E⁡[R2])2)\operatorname{Var}(A)=\pi^2\left(\operatorname{E}[R^4]-(\operatorname{E}[R^2])^2\right)

Compute

Compute

Use the uniform moment formula E[Rᵏ]=3ᵏ/(k+1).

E⁡[R2]=323=3\operatorname{E}[R^2]=\frac{3^2}{3}=3
E⁡[R4]=345=815\operatorname{E}[R^4]=\frac{3^4}{5}=\frac{81}{5}
Var⁡(A)=π2(815−9)=36π25\operatorname{Var}(A)=\pi^2\left(\frac{81}{5}-9\right)=\frac{36\pi^2}{5}

Answer

Answer

The area variance is 36pi squared divided by 5.

36π25(D)\boxed{\frac{36\pi^2}{5}\quad\text{(D)}}